How to Find the Period of a Function: Rules and Examples

The period of a function is a positive horizontal shift that makes its values repeat. For a nonconstant sine or cosine rule \(y=A\sin(Bx+C)+D\) or \(y=A\cos(Bx+C)+D\), the fundamental period is \(2\pi/|B|\). For tangent and cotangent, use \(\pi/|B|\). These formulas assume radians and nonzero \(A\) and \(B\).

What is a fundamental period?

A positive number \(P\) is a period when shifting every allowed input by \(P\) preserves the domain and gives \(f(x+P)=f(x)\). The fundamental period is the smallest positive period, when one exists. For example, \(\sin x\) repeats after \(2\pi\), \(4\pi\), and \(6\pi\); its fundamental period is \(2\pi\).

On a graph, measure the horizontal distance between corresponding points in consecutive cycles, such as adjacent peaks. A peak-to-trough distance is only half a cycle for a sine wave. The repeating variable may be an angle, distance or time; period is not always a time interval.

Not every function is periodic. For instance, \(f(x)=x\) never repeats after a positive horizontal shift. A constant function has every positive number as a period, so it has no smallest positive period.

Choose the correct period formula

• For \(\sin x\), \(\cos x\), \(\sec x\) and \(\csc x\), the base period is \(2\pi\).
• For \(\tan x\) and \(\cot x\), the base period is \(\pi\).
• In a nonconstant transformed rule \(A f(Bx+C)+D\), divide the base period by \(|B|\). Horizontal shifts and nonzero vertical scaling do not change the fundamental period.
• If the input is in degrees, use \(360^\circ/|B|\) for sine, cosine, secant and cosecant, and \(180^\circ/|B|\) for tangent and cotangent.

Read the coefficient of \(x\) after expanding the angle if needed. In \(\sin(3(x-2))\), that coefficient is \(3\). The sign of \(B\) changes orientation, while its absolute value determines the cycle length.

Periodicity does not require a graph to have a vertical line of symmetry, a domain of all real numbers or a bounded range. Tangent, for example, has undefined inputs and an unbounded range. For a sum or an expression such as \(|\sin x|\), check the whole rule: absolute value reduces the fundamental period of \(\sin x\) to \(\pi\). A common period of the parts of a sum is a period of the sum, but cancellation can produce a smaller one.

Worked examples: sine, cosine and tangent

1. Sine with a horizontal shift

For \(y=\sin(4x+5)\), the coefficient of \(x\) is \(4\). Thus \(P=2\pi/4=\pi/2\). The added \(5\) shifts the wave without changing the period.

2. Cosine with a vertical stretch

For \(y=9\cos(6x+4)\), use \(B=6\): \(P=2\pi/6=\pi/3\). The multiplier \(9\) changes amplitude, not the period.

3. Tangent with a negative coefficient

For \(y=\tan(-3x)\), tangent starts with base period \(\pi\). Divide by \(|-3|\) to get \(P=\pi/3\). Using the sine formula here would give a full repeat that is twice the fundamental period.

Exercises: find the fundamental period

Find the period of the function.

1. \(\color{blue}{y= tan3x + sin\frac{5x}{2}}\)
2. \(\color{blue}{y=sec(\pi x-2)}\)
3. \(\color{blue}{y=cot(-(\frac{2\pi}{3})x)}\)
4. \(\color{blue}{\:y=cos\left(-\left(\frac{2}{3}\right)x-\pi \right)}\)

1. \(\color{blue}{4\pi}\)
2. \(\color{blue}{2}\)
3. \(\color{blue}{\frac{3}{2}}\)
4. \(\color{blue}{3\pi}\)

Why these four answers work

1. For \(\tan(3x)+\sin(5x/2)\), the component periods are \(\pi/3\) and \(4\pi/5\). Their smallest positive common multiple is \(4\pi\): it is 12 tangent cycles and 5 sine cycles. The tangent poles require any period to be a multiple of \(\pi/3\); the remaining sine term then requires a multiple of \(4\pi/5\). This confirms \(4\pi\) is fundamental.
2. For \(\sec(\pi x-2)\), use the secant base period: \(2\pi/\pi=2\).
3. For \(\cot(-(2\pi/3)x)\), use the cotangent base period: \(\pi/(2\pi/3)=3/2\).
4. For \(\cos(-(2/3)x-\pi)\), use the cosine base period: \(2\pi/(2/3)=3\pi\).

Once you have the period, use it to place the key points in our guide to graphing the sine function.